Method of continuity

Let B be a Banach space, V a normed vector space, and ${\displaystyle (L_{t})_{t\in [0,1]}}$ a norm continuous family of bounded linear operators from B into V. Assume that there exists a constant C such that for every ${\displaystyle t\in [0,1]}$ and every ${\displaystyle x\in B}$

${\displaystyle ||x||_{B}\leq C||L_{t}(x)||_{V}.}$

Then ${\displaystyle L_{0}}$ is surjective if and only if ${\displaystyle L_{1}}$ is surjective as well.

The method of continuity is used in conjunction with a priori estimates to prove the existence of suitably regular solutions to elliptic partial differential equations.

We assume that ${\displaystyle L_{0}}$ is surjective and show that ${\displaystyle L_{1}}$ is surjective as well.

Subdividing the interval [0,1] we may assume that ${\displaystyle ||L_{0}-L_{1}||\leq 1/(3C)}$. Furthermore, the surjectivity of ${\displaystyle L_{0}}$ implies that V is isomorphic to B and thus a Banach space. The hypothesis implies that ${\displaystyle L_{1}(B)\subseteq V}$ is a closed subspace.

Assume that ${\displaystyle L_{1}(B)\subseteq V}$ is a proper subspace. Riesz's lemma shows that there exists a ${\displaystyle y\in V}$ such that ${\displaystyle ||y||_{V}\leq 1}$ and ${\displaystyle \mathrm {dist} (y,L_{1}(B))>2/3}$. Now ${\displaystyle y=L_{0}(x)}$ for some ${\displaystyle x\in B}$ and ${\displaystyle ||x||_{B}\leq C||y||_{V}}$ by the hypothesis. Therefore

${\displaystyle ||y-L_{1}(x)||_{V}=||(L_{0}-L_{1})(x)||_{V}\leq ||L_{0}-L_{1}||||x||_{B}\leq 1/3,}$

which is a contradiction since ${\displaystyle L_{1}(x)\in L_{1}(B)}$.

• Schauder estimates

• Gilbarg, D.; Trudinger, Neil (1983), Elliptic Partial Differential Equations of Second Order, New York: Springer, ISBN 3-540-41160-7